Should my fuel oil tank for my home furnace be vented? it only runs for a short time then shuts off.?

All oil tanks should be vented. It is very doubtful that is the reason that your oil burner runs for a while and shuts off.

Have you checked the filters? There are three. One in line, one inside the oil burner and another sinistered bronze incorporated into the injector.

When was the last time the injector was changed?

Is the igniter working? Can you see the flame? What is the color? is it even lighting? Is the air intake clean and free? I have seen them clogged up with pet hair.

Of course it should be vented. It IS vented, otherwise it couldn't be filled. You have some other problem. Have you ever changed the fuel oil filter? Best if you get a fuel oil furnace specialist to check. There is a control. There is a nozzle and flame detection. I'm curious about one thing. Have you been at the furnace when it shuts off? Does it run, then the fire goes out, and it runs another half minute with no flame, then shuts down? And you keep pushing the little red button on the control, to make it run some more?